47
47

Nov 19, 2015
11/15

by
Vic Christianto, Florentin Smarandache

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In the present article we argue that it is possible to find numerical solution of coupled magnetic resonance equation for describing wireless energy transmit, as discussed recently by Karalis (2006) and Kurs et al. (2007). The proposed approach may be found useful in order to understand the phenomena of magnetic resonance. Further observation is of course recommended in order to refute or verify this proposition.

Topics: numerical solution, equations

63
63

Nov 20, 2015
11/15

by
Florentin Smarandache

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In this short note we study the existence and number of solutions in the set of integers (Z) and in the set of natural numbers (N) of Diophantine equations of second degree.

Topics: natural numbers, Diophantine equations

62
62

Nov 19, 2015
11/15

by
Vic Christianto, Florentin Smarandache, Frank Lichtenberg

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It has been known for quite long time that the electrodynamics of Maxwell equations can be extended and generalized further into Proca equations. The implications of introducing Proca equations include an alternative description of superconductivity, via extending London equations. In the light of another paper suggesting that Maxwell equations can be written using quaternion numbers, then we discuss a plausible extension of Proca equation using biquaternion number. Further implications and...

Topics: Maxwell equations, Proca equation

32
32

Nov 20, 2015
11/15

by
Florentin Smarandache

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In the High School Algebra manual for grade IX (1981), pp. 103-104, is presented a method for solving systems of two homogenous equations of second degree, with two unknowns. In this article we’ll present another method of solving them.

Topics: homogenous equations, solving systems

37
37

Nov 20, 2015
11/15

by
Linfan Mao

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A natural behavior is used to characterize by differential equation established on human observations, which is assumed to be on one particle or one field complied with reproducibility. However, the multilateral property of a particle P and the mathematical consistence determine that such an understanding is only local, not the whole reality on P, which leads to a central thesis for knowing the nature, i.e. how to establish a physical equation with a proper interpretation on a thing. As it is...

Topics: human observations, non-solvable equations

42
42

Nov 20, 2015
11/15

by
Linfan Mao

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Applying this result, this paper discusses the →G-flow solutions on Schrodinger equation, Klein-Gordon equation and Dirac equation, i.e., the field equations of particles, bosons or fermions, answers previous questions by ”yes“, and establishes the many world interpretation of quantum mechanics of H. Everett by purely mathematics in logic, i.e., mathematical combinatorics.

Topics: G-flow, equations of particles

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51

Nov 18, 2015
11/15

by
Florentin Smarandache

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Six conjectures on pairs of consecutive primes are listed below together with examples in each case.

Topics: prime exponential equations, consecutive primes

110
110

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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Separable Equations We will now learn our �rst technique for solving differential equation. An equation is called separable when you can use algebra to separate the two variables, so that each is completely on one side of the equation. We illustrate with some examples.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Nonlinear...

Source: http://www.flooved.com/reader/1426

82
82

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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We know that a standard way of testing whether a set of n n-vectors are linearly independent is to see if the n _ n determinant having them as its rows or columns is non-zero. This is also an important method when the n-vectors are solutions to a system; the determinant is given a special name. (Again, we will assume n = 2, but the de�nitions and results generalize to any n.)

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1454

81
81

Nov 14, 2013
11/13

by
Vera Mikyoung Hur

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We give a comprehensive development of the theory of linear differential equations with constant coef�cients. We use the operator calculus to deduce the existence and uniqueness. We presents techniques for �nding a complete solution of the inhomogeneous equation from solu-tions of the homogeneous equation. We also give qualitative results on asymptotic stability.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1562

68
68

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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The purpose of these notes is to give some examples illustrating how naive numerical approximations to PDE's may not work at all as expected. In addition, the following two important notions are introduced: (I) von Neumann stability analysis - helps identify when (and if ) numerical schemes behave properly. (II) Artificial viscosity - a tool in stabilizing numerical schemes. These notes should be read in conjunction with the use of the MatLab scripts (in the Athena 18311-Toolkit at MIT) whose...

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1179

81
81

Nov 14, 2013
11/13

by
Jared Speck

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We would now like to derive an analog of d�Alembert�s formula in the physically relevant case of 1+3 dimensions. As we will see, the analogous formula, known as Kirchho_�s formula, can bederived_through the following steps. Given a solution u(t, x) to the 1 + 3 dimensional wave equation, we will de�ne a spherical_average of u centered at x. The average will depend on the averaging radius r. For �xed x, we will show that a slight modi�cation of the average will solve the 1 1...

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1601

71
71

Nov 14, 2013
11/13

by
Jared Speck

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As we will soon see, the PDE (1.0.1) has a unique solution verifying (1.0.2) as long as f(x) is su_ciently di_erentiable and decays su_ciently rapidly as |x|_ �. Much like in the case of the heat equation, we will be able to construct the solution using an object called the fundamental solution.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1614

74
74

Nov 14, 2013
11/13

by
Jared Speck

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We are now ready to derive Kirchho_�s famous formula.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1602

64
64

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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These notes give examples illustrating how conservation principles are used to obtain (phenomenological) continuum models for physical phenomena. The general principles are presented, with examples from traffic flow, river flows, granular flows, gas dynamics and diffusion.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1174

32
32

May 2, 2016
05/16

by
IASET JOURNALS

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In this paper some necessary and sufficient conditions of third order neutral differential equations are obtained to insure the oscillation of all solutions or converge to zero or tends to infinity as . Some examples are given to illustrate the obtained results.

Topics: Oscillation, Neutral Differential Equations

85
85

Nov 14, 2013
11/13

by
Hung Cheng

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Solutions expanded around an irregular singular point are distinctive in one aspect: they are usually in the form of an exponential function times a Frobenius series. Due to the factor of the exponential function, a solution near an irregular singular point behaves very differently from that near a regular singular point. It may blow up exponentially, or vanish exponentially, or oscillate wildly.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1338

81
81

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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These notes describe in some detail the continuum limit behavior of a very simple car following traffic flow model. The formation and behavior of shock waves is described. This model is the one solved by a set of MatLab scripts in the Athena 18311-Toolkit at MIT, which illustrate the phenomena described here. These are the scripts whose names end with the acronym CFSM.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1178

69
69

Nov 14, 2013
11/13

by
Jared Speck

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Let�s discuss how the wave equation arises as an approximation to the equations of �uid mechanics. For simplicity, let�s only discuss the case of 1 spatial dimension.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1600

90
90

Nov 14, 2013
11/13

by
Jared Speck

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In this course, we will mainly consider the case of free particles, in which V = 0 (i.e., the homogeneous Schr�odinger equation). In the case of free particles, there is an important family of solutions to (1.0.1), namely the free waves. The free wave solutions provide some important intuition about how solutions to the homogeneous Schr�odinger equation behave.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1607

86
86

Nov 14, 2013
11/13

by
Jared Speck

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In these notes, we introduce a class of evolution PDEs known as transport equations. Such equations arise in a physical context whenever a quantity is �transported� in a certain direction. Some important physical examples include the mass density �ow for an incompressible �uid, and the Boltzmann equation of kinetic theory. We discuss both linear transport equations and a famous nonlinear transport equation known as Burger�s equation. One of our major goals is to show that in contrast...

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1610

87
87

Nov 14, 2013
11/13

by
Jared Speck

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We will make use of an important mathematical operation called convolution.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1612

144
144

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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Definition: 1.1 The point z0 is called a branch point - for the complex (multiple) valued function f(z) - if the value of f(z) does not return to its initial value as a closed curve around the point is traced (starting from some arbitrary point on the curve), in such a way that f varies continuously as the path is traced.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1173

58
58

Nov 14, 2013
11/13

by
Jared Speck

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In this section, we will study (scalar-valued) functions _ on R^(1+n). They are sometimes called (scalar-valued) �elds on R^(1+n)

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1609

61
61

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In this paper we present three new examples of using the α-Discounting Multi-Criteria Decision Making Method in solving non-linear problems involving algebraic equations and inequalities in the decision process.

Topics: Multi-Criteria Decision Making Method, algebraic equations

105
105

Nov 14, 2013
11/13

by
Jared Speck

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We now derive our main representation formula for solution�s to Poisson�s equation on a domain _

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1616

100
100

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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When nonlinearities are "small" there are various ways one can exploit this fact - and the fact that the linearized problem can be solved exactly - to produce useful approximations to the solutions. We illustrate two of these techniques here, with examples from phase plane analysis: The Poincare-Lindstedt method and the (more exible) Two Timing method. This second method is a particular case of the Multiple Scales approximation technique, which is useful whenever the solution of a...

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1181

89
89

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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In two dimensions a Hopf bifurcation occurs as a Spiral Point switches from stable to unstable (or vice versa) and a periodic solution appears. There are, however, more details to the story than this: The fact that a critical point switches from stable to unstable spiral (or vice versa) alone does not guarantee that a periodic solution will arise, though one almost always does. Here we will explore these questions in some detail, using the method of multiple scales to find precise conditions...

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1176

480
480

Nov 16, 2015
11/15

by
Vic Chrisitianto, Florentin Smarandache

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Quaternion space and its respective Quaternion Relativity (it also may be called as Rotational Relativity) has been defined in a number of papers, and it can be shown that this new theory is capable to describe relativistic motion in elegant and straightforward way. Nonetheless there are subsequent theoretical developments which remains an open question, for instance to derive Maxwell equations in Q-space. Therefore the purpose of the present paper is to derive a consistent description of...

Topics: Quaternion space, Maxwell equations, Lorentz force.

60
60

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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Solitary waves are localized traveling steady pro�le solutions for dispersive nonlinear dynamical systems � usually modeled by a pde, or a system of pde�s. Thus, at least in 1+1 dimensions, they are relatively easy to characterize analytically � since they correspond to solutions of the ode�s to which the pde�s reduce in a coordinate moving with the wave.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1180

91
91

Nov 14, 2013
11/13

by
Jared Speck

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We now discuss a technique, known as separation of variables, that can be used to explicitly solve certain PDEs. It is especially useful in the study of linear PDEs. Although this technique is applicable to some important PDEs, it is unfortunately far from universally applicable.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1608

104
104

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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An equation for fundamental matrices: We have been saying �a� rather than �the� fundamental matrix since the system (1) doesn�t have a unique fundamental matrix: there are many ways to pick two independent solutions of x' = A x to form the columns of _.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1393

104
104

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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This notes give a few examples illustrating how continuum models can be derived from special limits of discrete models. Only the simplest cases are considered, illustrating some of the most basic ideas. These techniques are useful because continuum models are often much easier to deal with than discrete models with very many variables, both conceptually and computationally.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1175

82
82

Nov 14, 2013
11/13

by
Rodolfo R. Rosales

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Contents 1. Physical setup, assumptions, and notation. 2. Derivation of the governing equations. 3. Linearized governing equations.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1177

70
70

Nov 14, 2013
11/13

by
Jared Speck

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We now introduce a very important object called the energy-momentum tensor. As we will see, it encodes some very important conservation laws associated to solutions of (1.0.2).

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1603

84
84

Nov 14, 2013
11/13

by
Jared Speck

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We will now study some important properties of solutions to the heat equation �tu_D�u = 0. For simplicity, we sometimes only study the case of 1 + 1 spacetime dimensions, even though analogous properties are veri�ed in higher dimensions.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1617

17
17

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This flow in rivers is concerned with unsteady flow in open channel and it mathematically governed by the Saint Venant equation, using a four-point implicit finite difference scheme. For a one-dimensional applications, the relevant flow parameters are functions of time, and longitudinal positions. Considering the equations for the conservation of mass (continuity) and conservation of momentum. The mathematical method is empirical with the computer revolution, numerical methods are now effective...

Topics: Finite difference scheme, Numerical methods, Momentum equations.

76
76

Nov 14, 2013
11/13

by
Vera Mikyoung Hur

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An important result of mechanics is that a system of masses attached in (damped or undamped) springs is stable. A similar result is in network theory. In these notes, we study the differential equation of the form y" + py' + qy = f(t), where p, q are constants and f(t) represents the external forces.

Topics: Maths, Linear Algebra and Geometry, Differential Equations (ODEs & PDEs), Linear Algebra,...

Source: http://www.flooved.com/reader/1540

71
71

Nov 14, 2013
11/13

by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis

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Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1388

74
74

Nov 14, 2013
11/13

by
Jared Speck

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The results from the previous lecture produced one solution to the Dirichlet problem... But how do we know that this is the only one? In other words, we need to answer the uniqueness question (6) from the previous lecture. The next theorem addresses this question. We �rst need to introduce some important spacetime domains that will play a role in the analysis

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics

Source: http://www.flooved.com/reader/1611

31
31

Nov 22, 2016
11/16

by
Iaset Journal

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In such models the emphasis was on estimating and/or predicting the average value of Y conditional upon the fixed values of the X variables. But in many situations, such a one-way or unidirectional cause-and-effect relationship is not meaningful. This occurs if Y is determined by the X’s, and some of the X’s are, in turn, determined by Y. From that caused will be occurs problem in the estimation by use the OLS and the result not good to present the relationships between them, and we can't...

Topics: Econometrics, Simultaneous Equations, Market System

76
76

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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Remarks. Each of these three cases�one eigenvalue zero, pure imaginary eigenvalues, repeated real eigenvalue�has to be looked on as a borderline linear system: altering the coef�cients slightly can give it an entirely different geometric type, and in the �rst two cases, possibly alter its stability as well.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Linear...

Source: http://www.flooved.com/reader/1444

109
109

Nov 14, 2013
11/13

by
Matthew Hancock

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We now introduce the Fourier Transform and show how it is related to the solution of the Heat Problem on an in�nite domain.

Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...

Source: http://www.flooved.com/reader/1670

102
102

Nov 14, 2013
11/13

by
Richard Melrose

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Here is a slightly abbreviated version of what I did in lecture today on the completion of a normed space. The very last part I asked you to �nish as the �rst part of the second problem set, not due until February 24 due to the vagaries of the MIT calendar (but up later today). This problem may seem rather heavy sledding but if you can work through it all you will understand, before we get to it, the main sorts of arguments needed to prove most of the integrability results we will encounter...

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...

Source: http://www.flooved.com/reader/1585

59
59

Nov 14, 2013
11/13

by
Richard Melrose

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Theorem 8 (Baire). If M is a non-empty complete metric space and Cn _ M, n _ N, are closed subsets such that ... then at least one of the Cn�s has an interior point.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...

Source: http://www.flooved.com/reader/1573

78
78

Nov 14, 2013
11/13

by
Richard Melrose

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As a second �serious� application of at least the completeness part of the spectral theorem for self-adjoint compact operators, I want to discuss the Hermite basis for L2(R). Note that so far we have not found an explicit orthonormal basis on the whole real line, even though we know L2(R) to be separable, so we certainly know that such a basis exists. How to construct one explicitly and with some handy properties? One way is to simply orthonormalize � using Gramm-Schmidt � some...

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...

Source: http://www.flooved.com/reader/1582

75
75

Nov 14, 2013
11/13

by
Steven G. Johnson

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In these notes, written to accompany 18.06 lectures in Fall 2007, we discuss these mysteries: Fourier series come from taking concepts like eigenvalues and eigenvectors and Hermitian matrices and applying them to functions instead of �nite column vectors. In this way, we see that important properties like orthogonality of the Fourier series arises not by accident, but as a special case of a much more general fact, analogous to the fact that Hermitian matrices have orthogonal eigenvectors.

Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...

Source: http://www.flooved.com/reader/1672

75
75

Nov 14, 2013
11/13

by
Richard Melrose

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So, recall the de�nitions of a Lebesgue integrable function on the line (forming the linear space L^1(R)) and of a set of measure zero E _ R. The �rst thing we want to show is that the putative norm on L^1 does make sense.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...

Source: http://www.flooved.com/reader/1588

77
77

Nov 14, 2013
11/13

by
Richard Melrose

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I talked about step functions, then the covering lemmas which are the basis of the de�nition of Lebesgue measure � which we will do after the integral � then properties of monotone sequences of step functions.

Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...

Source: http://www.flooved.com/reader/1586

79
79

Nov 14, 2013
11/13

by
Vera Mikyoung Hur

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First-order linear differential equations. We will give a systematic method of solving �rst-order differential equations (of normal form) y' + p(x)y = f(x) on a given interval I, where p, f are continuous functions.

Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...

Source: http://www.flooved.com/reader/1551